The local isometry property (REF ), \(\Vert {\mathcal {M}_\text{i}^*}_{\Omega _\text{i}} {\mathcal {M}_\text{i}}_{\Omega _\text{i}}-\mathbf {I}_{\Omega _\text{i}}\Vert \!\le \!1/4\) , is proven in [1]}. In particular, it is obtained setting \(\text{Q}=\text{N}\)  and \(\delta =1/4\)  in [1]} and the upper bound holds with probability at least \(1-\text{N}^{-\alpha +1}\)  if \(\alpha \ge 1\)  and
\(\begin{split}\alpha \le \alpha _1 := &\min _{\text{i}\in \lbrace 1,2,\dots ,\text{R}\rbrace } \frac{1/16 \ (5/2 + 1/6)^{-1}}{\rho _{\mathbf {\Psi }_\text{i}}^{(\text{L}_\text{i})} \rho _{\mathbf {U}_\text{i}}^{(\text{S}_\text{i})} \log \left(2 \text{N}\text{L}_\text{i}\text{S}_\text{i}\right)} = \\ &\min _{\text{i}\in \lbrace 1,2,\dots ,\text{R}\rbrace } \frac{3}{128} \left(\rho _{\mathbf {\Psi }_\text{i}}^{(\text{L}_\text{i})} \rho _{\mathbf {U}_\text{i}}^{(\text{S}_\text{i})} \log \left(2 \text{N}\text{L}_\text{i}\text{S}_\text{i}\right)\right)^{-1}.\end{split}\) 
